The fact that the triangle inequality can be used in both directions in undirected graphs, makes it difficult to extend the lower bound constructions to undirected graphs, but it also seems to make more algorithmic tradeoffs possible for undirected than for directed graphs, as evident from the Cairo, Grossi, Rizzi [1]} algorithms. It thus seems possible that a better than 2 approximation algorithm running in linear time could be possible for undirected graphs.
